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The forced soliton equation is the starting point for semiclassical computations with solitons away from the small momentum transfer regime. This paper develops necessary analytical and numerical tools for analyzing solutions to the forced soliton equation in the context of two-dimensional models with kinks. Results include a finite degree of freedom regularization of soliton sector physics based on periodic and anti-periodic lattice models, a detailed analysis of numerical solutions, and the development of perturbation theory in the soliton momentum transfer to mass ratio Delta P/M. Numerical solutions at large transfer Delta P/M are capable of exhibiting, in a smooth and controlled fashion, extreme phenomena such as soliton-antisoliton pair creation and superluminal collective coordinate velocities, which we investigate.
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Fundamental competition of smooth and non-smooth bifurcations and their ghosts in vibro-impact pairs
Abstract A combined analysis of smooth and non-smooth bifurcations captures the interplay of different qualitative transitions in a canonical model of an impact pair, a forced capsule in which a ball moves freely between impacts on either end of the capsule. The analysis, generic for the impact pair context, is also relevant for applications. It is applied to a model of an inclined vibro-impact energy harvester device, where the energy is generated via impacts of the ball with a dielectric polymer on the capsule ends. While sequences of bifurcations have been studied extensively in single- degree-of-freedom impacting models, there are limited results for two-degree-of-freedom impacting systems such as the impact pair. Using an analytical characterization of impacting solutions and their stability based on the maps between impacts, we obtain sequences of period doubling and fold bifurcations together with grazing bifurcations, a particular focus here. Grazing occurs when a sequence of impacts on either end of the capsule are augmented by a zero-velocity impact, a transition that is fundamentally different from the smooth bifurcations that are instead characterized by eigenvalues of the local behavior. The combined analyses allow identification of bifurcations also on unstable or unphysical solutions branches, which we term ghost bifurcations. While these ghost bifurcations are not observed experimentally or via simple numerical integration of the model, nevertheless they can influence the birth or death of complex behaviors and additional grazing transitions, as confirmed by comparisons with the numerical results. The competition between the different bifurcations and their ghosts influences the parameter ranges for favorable energy output; thus, the analyses of bifurcation sequences yield important design information.
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- Award ID(s):
- 2009270
- PAR ID:
- 10385575
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Nonlinear Dynamics
- Volume:
- 111
- Issue:
- 7
- ISSN:
- 0924-090X
- Page Range / eLocation ID:
- p. 6129-6155
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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